Entropy, extremality, euclidean variations, and the equations of motion
By Xi Dong, Aitor Lewkowycz. 2017
159 citations recorded by the source corpus.
A bstractWe study the Euclidean gravitational path integral computing the Rényi entropy and analyze its behavior under small variations. We argue that, in Einstein gravity, the extremality condition can be understood from the variational principle at the level of the action, without having to solve explicitly the equations of motion. This set-up is then generalized to arbitrary theories of gravity, where we show that the respective entanglement entropy functional needs to be extremized. We also extend this result to all orders in Newton’s constant GN , providing a derivation of quantum extremality. Understanding quantum extremality for mixtures of states provides a generalization of the dual of the boundary modular Hamiltonian which is given by the bulk modular Hamiltonian plus the area operator, evaluated on the so-called modular extremal surface. This gives a bulk prescription for computing the relative entropies to all orders in GN . We also comment on how these ideas can be used to derive an integrated version of the equations of motion, linearized around arbitrary states.
Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/c3081053090a9275ff7003ad13fe0309b64c0a66
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In Atomic and Molecular Physics, and Optics
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Citations in the corpus
- Islands in Schwarzschild black holes (2020, background): “1) consists of two terms; the gravitational part3 of the generalized entropy, Sgravity, which is proportional to the total area (or volume for D > 4) of the boundaries of an island, ∂I, as [30–33]”
- Replica wormholes and the black hole interior (2019, methodology): “wheel geometry for nˇ1 Let’s begin by analyzing the pinwheel geometry in more detail, and working out its o-shell action near n= 1. The simple nal answer could be derived quickly from the methods in [7, 23], but we believe the following derivation may generalize better to integer n>1. We are inte”
- Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information (2020, background): “ the gravitational path integral (and in particular that this was implicit in earlier derivations of the quantum corrected Ryu-Takayanagi [50,51] and Hubeny-Rangamani-Takayanagi [52] entropy formulae [53,54]). While at this level the physical mechanisms behind such results remain somewhat mysterious”
- Bulk locality from modular flow (2017, background): “However, if we want to account for higher GN corrections, the results of [10] will receive order GN corrections and the two modular flows won’t be exactly the same [43].”